HYBRID ROBUST CONTROLLER BASED ON INTERVAL TYPE 2 FUZZY … · Lin and Chen (Lin & Chen, 2002) have...

S. Mohammadrezaei Nodeh et al.

106 Nexo Revista Científica / Vol. 32, No. 02, pp. 106-125 / Diciembre 2019

HYBRID ROBUST CONTROLLER BASED ON INTERVAL TYPE 2

FUZZY NEURAL NETWORK AND HIGHER ORDER SLIDING MODE

FOR ROBOTIC MANIPULATORS

CONTROLADOR HÍBRIDO ROBUSTO BASADO EN RED NEURONAL

FUZZY DE INTERVALO TIPO 2 Y MODO DESLIZANTE DE ALTO

ORDEN PARA ROBOTS MANIPULADORES

S. Mohammadrezaei Nodeh, M. H. Ghasemi*, H.R. Mohammadi Daniali

1Department of Mechanical Engineering, Babol Noshirvani University

of Technology, Mazandaran, Iran. *[email protected]

(recibido/received: 03-Agosto-2019; aceptado/accepted: 28-Octubre-2019)

ABSTRACT

Industrial arms should be able to perform their duties in environments where unpredictable conditions and

perturbations are present. In this paper, controlling a robotic manipulator is intended under significant

external perturbations and parametric uncertainties. Type-2 fuzzy logic is an appropriate choice in the face

of uncertain environments, for various reasons, including utilizing fuzzy membership functions. Also,

using the neural network (NN) can increase robustness of the controller. Although neural network does

not basically need to build its type-2 fuzzy rules, the initial rules based on sliding surface of higher order

sliding mode controller (HOSMC) can improve the system's performance. In addition, self-regulation

feature of the controller, which is based on the existence of the neural network in the central type-2 fuzzy

controller block, increases the robustness of the method even more. Effective performance of the proposed

controller (IT2FNN-HOSMC) is shown under various perturbations in numerical simulations.

Keywords: Type-2 Fuzzy Logic Neural Network; Higher-order Sliding Mode Control; Robotic Arm.

ABSTRACT

Los brazos industriale deben poder realizar sus tareas en entornos donde existen condiciones y

perturbaciones impredecibles. En este artículo, el control de un manipulador robótico está bajo

perturbaciones externas significativas e incertidumbres paramétricas. La lógica difusa de tipo 2 es una

opción adecuada frente a entornos inciertos, por varias razones, incluida la utilización de funciones de

membresía difusas. Además, el uso de la red neuronal (NN) puede aumentar la robustez del controlador.

Aunque la red neuronal no necesita básicamente construir sus reglas difusas tipo 2, las reglas iniciales

basadas en la superficie deslizante del controlador de modo deslizante de orden superior (HOSMC)

pueden mejorar el rendimiento del sistema. Además, la función de autorregulación del controlador, que se

basa en la existencia de la red neuronal en el bloque central del controlador difuso tipo 2, aumenta aún

Vol. 32, No. 02, pp. 106-125/Diciembre 2019

ISSN-E 1995-9516

Universidad Nacional de Ingeniería

COPYRIGHT © (UNI). TODOS LOS DERECHOS RESERVADOS

http://revistas.uni.edu.ni/index.php/Nexo

https://doi.org/10.5377/nexo.v32i02.9262



más la robustez del método. El rendimiento efectivo del controlador propuesto (IT2FNN-HOSMC) se

muestra bajo varias perturbaciones en simulaciones numéricas.

Palabras clave: Red neuronal de lógica difusa tipo 2; Control de modo deslizante de orden superior;

Brazo robótico.

1. INTRODUCTIÓN

Most of the robotic applications in real world have some level of uncertainties in their nature.

Uncertainties can be originated from several sources such as lack of the knowledge about the

environment, inaccurate and incomplete modelling and even receiving of controversy feedback and sensor

data. Fuzzy logic systems (FLS) is able to improve the capability of the system in facing uncertainties

since they are less dependent to system modelling. Also, knowledge of experts can be utilized in building

its linguistic rule base (Goguen, 1973; Zadeh, 1975). In general, two versions of fuzzy logic have been

introduced: type-1 fuzzy logic system (T1FLS) and type-2 fuzzy logic system (T2FLS). Both are widely

used in control methods when uncertainties are supposed to be taken into account (Mendel, 2004).

However, T1FLS membership grade for each input is a crisp number which can limit its capability to deal

with significant uncertainties. To address this weakness, T2FLS is introduced (Zadeh, 1975) in which

crisp numbers are replaced by intervals in membership functions. General type 2 fuzzy sets (GT2FS) and

interval type 2 fuzzy sets (IT2FS) are the two main categories which are developed from T2FLS (Nodeh,

Ghasemi, & Daniali, 2019). Various researches reveal that latter is more accurate and more efficient in

coping with uncertainties (Larguech, Aloui, Pagès, El Hajjaji, & Chaari, 2015; Park & Shin, 2018;

Ramesh, Panda, & Kumar, 2013; Shabaniniai, Etedali, & Ghadamyari, 2014; Zirkohi & Lin, 2015).

Another advantage of IT2FS is that fewer fuzzy rules are needed as it uses foot of uncertainty (FOU) in its

membership functions. In fact, by using FOU input/output domains can be covered by fewer membership

functions and it enables the technique to cover the uncertainties more effective. Smoother control signals,

better performance in stability control due to simplicity in building rule base are other benefits of the

IT2FS.

Sliding mode control (SMC) is a well-known robust method for nonlinear systems control in presence of

uncertainties and external perturbations (Hung, Gao, & Hung, 1993). Desirable transient response of the

system and insensitivity to perturbations and changes in system parameters are the main advantages of this

technique (Drakunov & Utkin, 1992). However, typical chattering which is caused by using switch

function is its main drawback. To address this issue, higher order sliding mode controller (HOSMC) is

developed by using higher order derivatives of classic SMC. In HOSMC sign function is exerted on higher

order derivatives, so accuracy is increased in discontinuous time sampling. One innovative method is to

Combine HOSMC with T2FLS. Type 2 fuzzy sliding mode control (T2FSMC) provides a robust

controller which is able to benefit from advantages of the both methods concurrently. For example, SMC

is able to reduce the degree of the system equations which leads to fewer rules to be needed by fuzzy logic

(Utkin, 2013). On the other hand, chattering effect of SMC can be reduced by using fuzzy mechanism

instead of switch function (Song & Smith, 2006).

Generally, there are two methods for combining sliding mode and fuzzy logic: employing fuzzy logic in a

sliding mode controller (SMC is the main controller) and employing sliding mode in a fuzzy controller

(FLC is the main controller). There exist advantages and disadvantages for each of these combinations. In

the first method, the employed switch function might be classic sliding mode and the type-II fuzzy logic

connects various control algorithms as a unique system. For 2D systems, sliding line is a function of error

and error derivative which is a 1D function. Thus, less rules are required to approximate the 1D function

which shows how this combination can reduce size of rule base. In the second method, a nonlinear sliding

mode is generated which has an advantage over classic sliding mode which uses a linear sliding surface.

Although in order to improve performance of the controller, width of the sliding surface is an important

parameter. Shortcoming of this method lies in the necessity of defining control parameters of the sliding



mode accurately by an expert. What is considered in this paper is using the second method for controlling

a robotic operator in the presence of uncertainty and external disturbance in which IT2FS is the main

controller and HOSM block is the regulator.

Lin and Chen (Lin & Chen, 2002) have used fuzzy-sliding mode method to control a class of uncertain

nonlinear MIMO systems. In this method, fuzzy-sliding mode controller has been used to approximate

equivalent control in neighborhood of the sliding manifold. They have showed that the system is able to

cover parametric uncertainty and compensate external disturbance. The method has been implemented on

a double-link arm and its stability is proved using Lyapunov method. Hu and Woo (Hu & Woo, 2006)

have proposed a method combining sliding mode and neural network with different weights where the

weights are determined using a fuzzy controller. Since equations are highly nonlinear, coupled and time-

variant, robotic arms are affected by various uncertainties. Although sliding mode control overcomes

uncertainties and fast transient response, it results in chattering due to discontinuity of the signal. Thus,

they have combined fuzzy-robust-neural network methods to overcome the challenges. They have proved

convergence and stability of the system using direct Lyapunov method. Simulations under various

conditions show proper performance of the controller.

Medhaffar et al. (Medhaffar, Derbel, & Damak, 2006) have proposed a decoupled sliding mode-fuzzy

controller for a robotic arm. This controller has been proposed to control a group of MIMO systems with

nonlinear dynamic. In fact, a fuzzy method has been used to approximate nonlinear functions of the

system. In addition, local controllers have been used to decouple equations. Finally, the method has been

implemented on a robotic arm with two degrees of freedom to control the implementation trajectory.

Onder (Efe, 2008) has presented a fuzzy-sliding mode controller for a double-link arm. Parameters of the

controller have been adjusted such that the system under control has a limited tracking error. In this

method, an adaptive process has been used to improve the controller based on integration of the feedback

signal.

Huang et.al. (Huang, Kuo, & Chang, 2008) have studied control of nonlinear systems in the presence of

uncertainties using fuzzy-sliding mode method. An adaptive adjustment method without high frequency

switch has been employed to handle bounded uncertainties. Robustness and stability of the system has

been proved using Lyapunov theory. However, during control process, it has been observed that upper

bound of uncertainty does not need to be known. The proposed method has been verified by experimental

tests. Roopaei et.al. (Roopaei, Zolghadri, & Meshksar, 2009) have proposed a fuzzy-sliding mode

controller to compensate uncertainty and disturbance. They have used two fuzzy systems to complete the

sliding mode block. In addition, they have used fuzzy logic to approximate some parts of the system. in

order to overcome chattering, fuzzy model has been used like saturation function. Its advantage is that it

does not require to know uncertainty and disturbance bounds. They have proved stability and convergence

of the closed loop system using Lyapunov theory and Burbalat lemma. Noroozi et.al. (Noroozi, Roopaei,

& Jahromi, 2009) have proposed a fuzzy-sliding mode controller for a class of nonlinear systems in the

presence of uncertainty and external disturbance. Their innovation is based on recognition of a part of the

system and not requiring to know uncertainty and disturbance bounds. In addition, chattering has been

eliminated without damaging robustness of the controller.

Hacioglu et.al. (Hacioglu, Arslan, & Yagiz, 2011) have studied controlling a double-arm with two degrees

of freedom while moving objects. Since these robots are mainly used in dangerous applications like

transportation of radioactive material and explosive disposal, they applied a sliding mode controller

lacking improved chattering by the fuzzy logic block to the robot so that it can scan the desired trajectory

with high accuracy and security. In order to evaluate performance of the controller against parametric

changes and external disturbances, results of sudden load changes and noise have been presented. Amer

et.al. (Amer, Sallam, & Elawady, 2011) have used fuzzy-sliding mode method to control a robotic arm.

They have integrated sliding surface adjusted by PID signal to prevent chattering. In addition, they have

adjusted output gain of the controller by a separate fuzzy block online. System stability has been proved

using Lyapunov method. They have implemented their method on a robotic arm with 3 degrees of

freedom in the presence of uncertainty and showed its positive effect of tracking. Nekoukar and Erfanian



(Nekoukar & Erfanian, 2011) have proposed an adaptive learning algorithm and a fuzzy logic system to

estimate dynamic system for controlling a robotic operator using sliding mode method. Advantage of this

method is that initial dynamic of the robot is not required to be known which results in stability and time-

limited convergence of the proposed algorithm. Cerman and Husek (Cerman & Hušek, 2012) have

proposed a fuzzy-sliding mode controller for a class of nonlinear continuous systems with unknown

dynamic and bounded disturbance. Their main idea includes a self-adjusting fuzzy mechanism for

adjusting sliding mode parameters and changing gains. This modification reduced chattering and increased

convergence rate. They implemented the proposed controller on hydroelectric servomotors. Kayacan and

Kaynak (Kayacan & Kaynak, 2012) have combined sliding mode and IT2FS and used neural network

with triangular membership function to control speed of a servomotor. Instead of minimizing an error

function, they have adjusted weights of the neural network such that balance equation is held.

Niknam et al. (Niknam, Khooban, Kavousifard, & Soltanpour, 2014) have proposed an optimal controller

by combining type-II fuzzy logic and sliding mode for controlling a specific class of nonlinear systems.

To this end, particle swarm optimization algorithm has been employed for adjusting parameters of input

and output membership functions. Despite proper performance of the algorithm, dynamic system of SISO

and its equations are limited. Keymasi and Moosavian (Khalaji & Moosavian, 2014) have employed two

kinematic and dynamic controllers to stabilize motion of a mobile wheeled robot with a trailer around time

trajectories. They have employed kinematic control of output feedback and fuzzy sliding mode control.

They have used practical simulation and implementation to illustrate performance of their algorithm.

Soltanpour et al. (Soltanpour, Khooban, & Khalghani, 2016) have approximated boundary layer of system

uncertainties to overcome chattering of the sliding mode method. Before that, they have reduced order of

system uncertainties using linearizer feedback. The method of interest has been applied to a reverse

pendulum. However, employing this method is limited to nonlinear system of the problem. Hendel et al.

(Hendel, Khaber, & Essounbouli, 2015) have used HOSM to reduce chattering of an uncertain system and

approximated uncertain part of the system using type-II fuzzy logic. The simulated the method on a

system with two state variables. Naik et al. (Naik, Samantaray, Roy, & Pattanayak, 2015) have compared

fuzzy-PD controller and sliding mode for a double-link arm. In addition, they have used model-based and

model-less methods for both controllers. Finally, the designed fuzzy-sliding mode controller has

advantages of both techniques. Simulations showed positive performance of the controller and acceptable

tracking error. Farahmand et al. (Farahmand, Ghasemi, & Salari, 2018) have used a hybrid nonlinear

fuzzy-sliding mode controller in a flying robot. They have reduced chattering of the controller compared

to its classic counterpart by using fuzzy logic as the external loop of the controller and spiral sliding mode

in the internal loop. Simulations have been performed in Simulink.

Camci et al. (Camci, Kripalani, Ma, Kayacan, & Khanesar, 2018) have employed type-II fuzzy logic

equipped with neural network to control a flying robot. First order sliding mode optimized using PSO has

been used to adjust parameters of the neural network. They have compared their simulation results with

practical values of the PID controller. Li et al. (Li, Wang, Wu, Lam, & Gao, 2017) have proposed a

controller based on optimal sliding mode for main IT2FS block. Integral sliding surface has helped

neutralizing disturbance and minimizing control error. Despite proper performance of the controller in

simulating the reverse pendulum, chattering of the system is significant. Nafia et al. (Nafia, El Kari, Ayad,

& Mjahed, 2018) have proposed a two-step controller for controlling n-link robotic operator. First, an

adaptive type-II fuzzy logic block has been used to approximate uncertainty of a system. Then, a hybrid

type-II fuzzy-sliding mode controller has been used for control in the presence of uncertainty and external

disturbance. The simulations have been done on a two-link robotic manipulator. In this work, a T2FLS

controller is developed in which the rule base acts in terms of second order SMC inputs and then neural

network calculation is used to determine final control signal. The proposed controller is intended to be

worked on a robotic arm with parametric uncertainties in presence of external perturbations. For this aim,

the following factors must be met by the proposed controller:

- Robustness of the system against parametric uncertainties of the manipulator and external

perturbations.



- Ensure the efficiency of the T2FLcontroller in limited time.

- No need to complicated offline computation.

To do so, an online regulation algorithm is needed. Although IT2NN (neural network based on type-2

fuzzy logic) is capable to handle control task in presence of uncertainty, it is not able to cover external

perturbations with any domain. To improve the controller, HOSMC outputs are employed as an input for

the main IT2NN controller. The organization of this article is as follows. In section 2, dynamic model of a

manipulator is presented. In the next Section, the control method is described. In Section 4, stability

analysis of the controller is investigated. Finally, section 5 is results and discussions which reveals the

advantages of the proposed method in numerical simulations.

2. DYNAMIC MODEL OF A ROBOT MANIPULATOR

For an n-link planar rigid manipulator, the Euler-Lagrange dynamic equations can be written as

𝐻(𝑞)�� + 𝑍(𝑞. ��) = 𝜏 (1)

where angular states of joints are considered as system states in which 𝑞. ��. ��𝜖𝑅𝑛 are joint position,

velocity and acceleration vectors respectively, 𝐻(𝑞) is positive definite inertia matrix, 𝜏 is the control

vector representing the control torque, and 𝑍(𝑞. ��) is matrix of system nonlinear expressions matrix.

Equation (1) can be rewritten as follow:

𝐻(𝑞)�� + 𝐶(𝑞. ��)�� + 𝑈𝑑�� + 𝑈𝑠(��) + 𝜏𝑑(𝑞. ��) + 𝑔(𝑞) = 𝜏 (2)

Nonlinear terms are Coriolis accelerations matrix, the dynamic coefficient matrix, the static friction

vector, the matrix of perturbation vector and un-modelled dynamics, and gravitational vector which are

defined by 𝐶(𝑞. ��), 𝑈𝑑𝜖𝑅𝑛×𝑛, 𝑈𝑠(��)𝜖𝑅

𝑛, 𝜏𝑑(𝑞. ��), and 𝑔(𝑞), respectively. It should be noted that the

only constraint on 𝜏𝑑(𝑞. ��) is its compatibility with the dimensions and the weight of the robot. For

instance, a light robotic arm is expected to withstand some perturbations in an appropriate domain, while a

heavy robot has easier job. In equation (2), there exist highly nonlinear terms such as trigonometric multiplications and it is also assumed that has parametric uncertainties which means the mass and inertia

of the links are unknown.

3. CONTROL METHOD

In this paper, goal of control method is to construct a control loop in which desired trajectory of 𝑞𝑑𝜖𝑅𝑛 in

presence of parametric uncertainty and external perturbation is followed. Moreover, the proposed

controller should not need offline complicated calculations. To design such a controller, first mathematical

equations of the neural network block based on type 2 fuzzy logic as the main control block is developed.

Then, equations of the SMC as the outer loop are obtained. Finally, the complete controlling system and

its block diagram is illustrated.

3.1 Neural Network Block based on Type 2 Fuzzy Logic

In this section, some terms and definitions of type 2 fuzzy sets are discussed. A fuzzy set of type-2 can be

expressed by the following form:

𝐴 = ∫𝜇𝐴(𝑥)

𝑥

𝑥∈𝑋

(3)



where, 𝑋 is reference set, 𝑥 is initial variable or primary variable. 𝜇𝐴(𝑥) is secondary membership

function. 𝐽𝑥and 𝑓𝑥(𝑢) are set of initial membership grades and secondary membership grade respectively.

The type-2 fuzzy sets is described by Zadeh [22] according to (5) and (6):

(4) 𝐴 =∪∀𝑥∈𝑋 𝜇𝐴(𝑥)

(5)

𝜇𝐴(𝑥) = ∫𝜇𝐴(𝑥. 𝑢)

(𝑢)

𝑢∈𝐽𝑥

; 𝜇𝐴(𝑥. 𝑢) = 𝑊𝑥𝑖 .

0 ≤ 𝑊𝑥𝑖 ≤ 1 𝑖 = 1.2. … . 𝑁

where, 𝜇𝐴(𝑥) is the membership function and 𝑊𝑥𝑖 are secondary variable weights.

The better performance of the T2FLS in comparison to conventional FLS, especially when there are high

levels of uncertainty and nonlinear dynamics, has been proven in various studies (Hamza, Yap, &

Choudhury, 2017). However, because of the type reduction processes such as Karnink-Mendel algorithm

(KM), computational cost of the technique is a drawback. Using neural network (NN) is one possible

solution to reduce computational cost (Zeghlache, Saigaa, & Kara, 2016). In fact, NN is an efficient

technique to estimate controlling parameters without previous knowledge about them and its dynamic

system. Structure of an interval type 2 fuzzy neural network (IT2FNN) is illustrated in Fig. 1. As it is

depicted, IT2FNN has 5 different levels as follows:

Level 1: input level in which input variables of IT2FNN are defined.

Level 2: membership level in which each node represents a membership function. Foot of uncertainty is

defined by upper membership function (UMF) and lower membership function (LMF) in equations (6)

and (7), respectively.

𝜇(𝑥𝑖) =

{

𝑁(𝜉. 𝜎𝑖𝑗. 𝑥𝑖) . 𝑥𝑖 < 𝜉

1. 𝜉 ≤ 𝑥𝑖 < 𝜉

𝑁(𝜉. 𝜎𝑖𝑗. 𝑥𝑖). 𝑥𝑖 < 𝜉

(6)

𝜇(𝑥𝑖) =

{

𝑁(𝜉. 𝜎𝑖

𝑗. 𝑥𝑖) . 𝑥𝑗 ≤

𝜉 + 𝜉

2

𝑁(𝜉. 𝜎𝑖𝑗. 𝑥𝑖) . 𝑥𝑗 >

𝜉 + 𝜉

2

(7)

where, 𝜎𝑖𝑗 and ξ are standard deviation and gaussian average of the i-th input, respectively.

Level 3: rule base level in which each node corresponds to a fuzzy rule and is used through fuzzy operator

to calculate (8) variable:

𝐹𝑖 = [∏𝜇𝑗𝑖

𝑛

j=1

∏𝜇𝑗𝑖

𝑛

𝑗=1

] = [𝑓𝑖 . 𝑓𝑖]

(8)

Level 4: type reduction level in which the output is calculated by assuming [𝜔𝑙𝑗, 𝜔𝑟

𝑗] as geometric center

of fuzzy type 2 set. KM algorithm is represented in equations. (9) and (10).



Figure 1. Structure of an interval type 2 fuzzy neural network (IT2FNN)

𝑦𝑙 =∑ 𝑓𝑗𝐿𝑗=1 𝑤𝑙

𝑗+ ∑ 𝑓𝑗𝑀

𝑗=𝐿+1 𝑤𝑙𝑗

∑ 𝑓𝑗𝐿𝑗=1 + ∑ 𝑓𝑗𝑀

𝑗=𝐿+1

= 𝑊𝑙𝑇𝑔𝑙 (9)

𝑦𝑟 =∑ 𝑓𝑗𝑅𝑗=1 𝑤𝑟

𝑗+ ∑ 𝑓𝑗𝑀

𝑗=𝑅+1 𝑤𝑟𝑗

∑ 𝑓𝑗𝑅𝑗=1 + ∑ 𝑓𝑗𝑀

𝑗=𝑅+1

= 𝑊𝑟𝑇𝑔𝑟

(10)

where, 𝑊𝑟 = [𝜔𝑟1. … . 𝜔𝑟

𝑀]𝑇, 𝑊𝑙 = [𝜔𝑙1. … . 𝜔𝑙

𝑀]𝑇. 𝑔𝑙 and 𝑔𝑟 are defined in equations (11) and (12):

𝑔𝑙 = [𝑓𝑙1

∑ 𝑓𝑙𝑖𝑀

𝑖=1

. …. 𝑓𝑙𝑀

∑ 𝑓𝑙𝑖𝑀

𝑖=1

]𝑇 (11)

𝑔𝑟 = [𝑓𝑟1

∑ 𝑓𝑟𝑖𝑀

𝑖=1

. …. 𝑓𝑟𝑀

∑ 𝑓𝑟𝑖𝑀

𝑖=1

]𝑇 (12)

Level 5: output level in where each output acts as a defuzzifier and the final output is in the following

form:

𝑦 =𝑦𝑙 + 𝑦𝑟2

=𝑊𝑙

T𝑔𝑙 +𝑊𝑟T𝑔𝑟

2 (13)

In general, this structure does not need predetermined rules as it is able to use online learning based on

inputs and outputs of the controlling system. However, using expert’s knowledge in building the initial

rule base can be helpful. This rule base can be based on output of a HOSMC which is explained briefly in

the next section.

3.2 Higher Order Sliding Mode Control

The classic sliding mode controller design consists of two levels of designing the slip surface and

selecting the control rule for tracking this surface. By calculating slip surface, closed loop system order is

reduced and a robust behaviour for robot is provided.

A simple system can be considered as (14):

𝑥(𝑛)(𝑡) = ℎ(𝑥) + 𝑝(𝑥)𝑢 (14)

where, 𝑥 and 𝑢 are state matrix and control input, respectively. ℎ(𝑥) is a nonlinear indefinite function?

sign and bounds of 𝑝(𝑥) are determined by a continuous function of 𝑥.

Inputs Membership

Functions Rule

base Type

Reduction Output



Here, the problem is that in limited time, following desired variable state must be followed by the state 𝑥.

(15) 𝒙𝒅 = [𝑥𝑑 𝑥�� … 𝑥𝑑(𝑛−1)]𝑇

Despite ℎ(𝑥) and 𝑝(𝑥) are uncertain, finite control input 𝑢 and initial conditions is used as:

(16) 𝒙𝒅(0) = 𝒙(0)

Tracking error is defined as:

(17) 𝑒 = 𝒙 − 𝒙𝒅 = [𝑒 𝑒 … 𝑒(𝑛−1)]𝑇

A high speed switching mechanism is necessary to ensure the desired performance of a closed loop

system. Because of the practical limitations, direct implementing of the abovementioned controlling

algorithm leads the system to oscillation in the neighbourhood of the sliding surface. To reduce the typical

chattering effect in this paper, HOSMC is developed from original SMC. By using higher order

derivatives, HOSMC provides a more robust controller with smoother signals. Second order SMC model

is represented in equations (18) to (20):

(18) 𝑆(𝑥. 𝑡) = (

𝑑

𝑑𝑡+ 𝜆)

𝑛−1

𝑒 ; 𝑛 = 3 → 𝑠 = 𝐾𝐷�� + 𝐾𝑃𝑒 + 𝐾𝐹��

(19) ��𝑒𝑞 = (𝐾𝐷��)−1[𝐾𝑃�� + 𝐾𝐹𝑒 + 𝐾𝐷ℎ�� + 𝐾𝐷��𝑔(𝑞) + 𝐾𝐷��𝑑]

(20) 𝑢 = ��[��𝑒𝑞 − 𝐾𝑠𝑎𝑡 (𝑠

𝜙)]

Where 𝑆 is sliding variable, 𝜆 is a positive constant, 𝐾𝐷, 𝐾𝑃 and 𝐾𝐹 are PID parameters for sliding surface,

and K is paramer for calculating uncertainty effect on the system. 𝜙 is boundry layer width in which

chattering relaxation can have occurred.

Equation (18) is a general form for writing sliding surface with different orders. However, it is common to

assign a different tuning parameter to each obtained signal which can be tuned separately. The control rule

(20), which is the result of a second-order sliding mode and a saturation switch function, is able to delete

the chattering of signals effectively. The main reason behind the chattering removing performance of the

HOSMC is using of higher derivatives which leads to smoother signals.

3.3 The proposed controlling system

In Fig. 2 the block diagram of the proposed control is depicted. The sliding surface is designed based on

equation (29). In this diagram, the input is determined by NN using output and reference signals based on

T2FLC and HOSMC. External perturbation is represented by 𝜏𝑑.

Inputs of the T2FLC are 𝑠 and ��. Input and output signals of the controller have 5 type 2 fuzzy sets which

are called negative big (NB), negative small (NS), zero (ZE), positive small (PS), and positive big (PB).

Input range is determined by estimation of s in the presence of perturbations. This range is assumed to be

[-3,3] in simulations.



Figure 2. Flow-chart of IT2FNN-HOSMC

Membership functions are considered in Gaussian form. Fuzzy rules are defined based on analysis of

system trajectories towards sliding surface. For example, if the system states are far from the sliding

surface and its velocity is small, a big input is needed to get closer to the sliding surface. The fuzzy rule

base is represented in table 1.

Table 1. Fuzzy controller rule base

PB PS ZE NS NB s s

NB NB NB NM ZE PB

NB NB NM ZE PM PS

NB NM ZE PM PB ZE

NM ZE PM PB PB NS

ZE PM PB PB PB NB

In Fig. 3, mapping of error of the system states and its derivative to the sliding surface is illustrated. Then,

output of the controller will be calculated by T2FLC and using fuzzy rules with defined membership

functions, based on the rule base in Table. 1.

Figure 3. Mapping of error of the system states and its derivative to the sliding surface

Membership

Functions



4. STABILITY ANALYSIS

By making assumption that 𝑂𝐹 is output of fuzzy controller and 𝑂𝑁𝑁 is output of the neural network

block, stability analysis of the system is carried out by the following positive definite Lyapunov function:

(21) 𝑉 =1

2𝑂𝐹2(𝑡)

(22) �� = 𝑂𝐹��𝐹 = 𝑂𝐹(��𝑁𝑁 + ��)

Since the output of the NN is defined in the following form:

(23)

𝑂𝑁𝑁 = 𝑞 ∑ ∑ 𝑓𝑖𝑗

𝐽𝑗=1

𝐼𝑖=1 𝑊𝑖𝑗

∑ ∑ 𝑊𝑖𝑗𝐽𝑗=1

𝐼𝑖=1

+(1 − 𝑞)∑ ∑ 𝑓𝑖𝑗

𝐽𝑗=1

𝐼𝑖=1 𝑊𝑖𝑗

∑ ∑ 𝑊𝑖𝑗𝐽𝑗=1

𝐼𝑖=1

= 𝑞∑∑𝑓𝑖𝑗

𝐽

𝑗=1

𝐼

𝑖=1

𝑊𝑖�� + (1 − 𝑞)∑∑𝑓𝑖𝑗

𝐽

𝑗=1

𝐼

𝑖=1

𝑊𝑖𝑗

(24) ��𝑁𝑁 = 𝑞∑∑(

𝐽

𝑗=1

𝐼

𝑖=1

𝑓𝑖��𝑊𝑖�� + 𝑓𝑖𝑗𝑊𝑖�� ) + (1 − 𝑞)∑∑(

𝐽

𝑗=1

𝐼

𝑖=1

𝑓𝑖��𝑊𝑖𝑗 + 𝑓𝑖𝑗

𝑑

𝑑𝑡𝑊𝑖𝑗)

Therefore,

(25) �� = 𝑂𝐹(∑∑(

𝐽

𝑗=1

𝐼

𝑖=1

𝑓𝑖��𝑊𝑖�� + 𝑓𝑖𝑗𝑊𝑖�� ) + (1 − 𝑞)∑∑(

𝐽

𝑗=1

𝐼

𝑖=1

𝑓𝑖��𝑊𝑖𝑗 + 𝑓𝑖𝑗

𝑑

𝑑𝑡𝑊𝑖𝑗) + ��)

Because derivatives of the 𝑊𝑖�� and 𝑊𝑖𝑗

equal to zero, derivative of the Lyapunov function can be

expressed as follows:

(26) �� = 𝑂𝐹(∑∑𝑓𝑖��

𝐽

𝑗=1

𝐼

𝑖=1

(𝑞𝑊𝑖�� + (1 − 𝑞)𝑊𝑖𝑗) + ��)

By defining 𝑓𝑖�� in equation (38),

(27) 𝑓��𝑗 = −𝑞𝑊𝑖��+(1−𝑞)𝑊𝑖𝑗

)

(𝑞��+(1−𝑞)��)𝑇(𝑞��+(1−𝑞)��)𝛼𝑠𝑔𝑛(𝑂𝐹)

where, 𝛿 is a positive number which is bigger than positive number 𝜂. By substituting equation (27) in

(26), the following equation can be obtained:

(28) �� = τ(−𝛿𝑠𝑔𝑛(𝑂𝐹) + ��) < (−𝛿|𝑂𝐹| + η|𝑂𝐹|) < 0

5. RESULTS AND DISCUSSIONS

In this section, the performance of the proposed algorithm is inspected by carrying out simulations on a 2

DOF rigid two links manipulator.

The proposed IT2FNN-HOSMC algorithm is used to control a rigid two links manipulator. In this case

mass and length of the links are specified as 𝑚1 = 𝑚2 = 1 𝐾𝑔 and 𝑙1 = 𝑙2 = 0.5 𝑚, respectively. The

dynamical effects such as static frictions and nonlinear viscous are determined by equations (29) and (30)

[26].



𝐹𝑑�� = [

𝑓𝑑��1𝑓𝑑��2𝑓𝑑��3

] = [

5��15��25��3

] (29)

𝐹𝑠(��) = [

𝑓𝑠𝑠𝑖𝑔𝑛(��1)𝑓𝑠𝑠𝑖𝑔𝑛(��2)𝑓𝑠𝑠𝑖𝑔𝑛(��3)

] = [

5𝑠𝑖𝑔𝑛(��1)5𝑠𝑖𝑔𝑛(��2)5𝑠𝑖𝑔𝑛(��3)

] (30)

To assess the performance of the proposed approach, two sets of tracking trajectory simulations are

performed. In each set, different type of external perturbations is exerted. In the first set, the desired

trajectory is 𝑞𝑑(𝑡) = 1 − cos (𝜋𝑡) which is a periodic signal for joint angle. To make the situation more

challenging for the controller, initial position of the desired joints is set as 2 radians. Control parameters

for classic SMC and second-order SMC (PID sliding surface) are 𝐾𝑃 = 𝑑𝑖𝑎𝑔{300. 300. 300}. 𝐾𝐼 =𝑑𝑖𝑎𝑔{250. 250. 250}. and 𝐾𝐷 = 𝑑𝑖𝑎𝑔{20. 20. 20}. In the first set, perturbation is considered as a sinusoidal wave with specified amplitude (Fig. 4). In

equation (31), the frequency of this wave is selected small enough for conventional controllers such as

SMC to not be unable to control the manipulator properly. In all simulations, the performance of the

proposed controller is compared to conventional SMC. The performance of the two methods in tracking

problem of the first and second link is shown in Figures 5 and 6, respectively. The proposed technique is

able to converge effectively in an acceptable time period of less than 1 second for the both links, whereas

conventional SMC is able to do so in a much longer period of time. In fact, delay which is a typical

problem for conventional SMC has been addressed by implementing HOSMC as input block for the main

controller. It should be noted that the reason of unsuitable performance of the SMC is that the frequency

of the perturbations is deliberately selected as small as it could cause problem for this method. In Figs. 7

and 8, the controlling torque needed for tracking simulation is shown. As soon as SMC converge to the

desired path, remarkable chattering effect can be seen for this technique. On the other hand, IT2FNN-

HOSMC by using IT2FS in it is able to reduce the chattering effect. It should be noted that the range of

the torque of each motor is constrained to [-20,20] which is the main reason of chattering effect in the

proposed technique. By using stronger motor, even this amount of chattering can be vanished. Moreover,

neural network helps the proposed method to maintain control input in desired interval.

In the last simulations for this part, position errors of first and second joints are shown in Figures 9 and 10,

respectively. It is obvious that the proposed method is able to converge to zero and keep the position error

in both joints in an acceptable range much faster than SMC (Fig. 11).

𝜏𝑑 = [

20 + 20 𝑠𝑖𝑛(20(𝑡 − 1)) + 30 𝑠𝑖𝑛(10(𝑡 − 0.5))

20 + 20 𝑠𝑖𝑛(20(𝑡 − 1)) + 30 𝑠𝑖𝑛(10(𝑡 − 0.5))

20 + 20 𝑠𝑖𝑛(20(𝑡 − 1)) + 30 𝑠𝑖𝑛(10(𝑡 − 0.5))

+20𝑢(𝑡 − 0.5) + 20𝑢(𝑡 − 1)

+20𝑢(𝑡 − 0.5) + 20𝑢(𝑡 − 1)

+20𝑢(𝑡 − 0.5) + 20𝑢(𝑡 − 1)

] (31)

Figure 4. Sinusoidal wave perturbation profile



Figure 5. Comparison between SMC and IT2FNN-HOSMC in trajectory tracking of joint 1 under sinusoidal wave

perturbation

Figure 6. Comparison between SMC and IT2FNN-HOSMC in trajectory tracking of joint 2 under sinusoidal wave

perturbation

Figure 7. Chattering effect in joint 1 by using SMC and IT2FNN-HOSMC under sinusoidal wave perturbation



Figure 8. Chattering effect in joint 2 by using SMC and IT2FNN-HOSMC under sinusoidal wave perturbation

Figure 9. Position error in joint 1 by using SMC and IT2FNN-HOSMC under sinusoidal wave perturbation

Figure 10. Position error in joint 2 by using SMC and IT2FNN-HOSMC under sinusoidal wave perturbation



Figure 11. Sliding Variable for first perturbation (IT2FNN-HOSMC)

In the second set of the simulations, random perturbations are exerted on both joints (Fig. 12). Controlling

robotic systems in the presence of this type of perturbation is always challenging. To simulate random

perturbation here, a random function is employed. The frequency of perturbation is as small as it could

cause chattering issue for conventional SMC.

Figures 13 and 14 show comparison between performance of SMC and proposed technique in trajectory

tracking problem for first and second joints of the manipulator, respectively. As it is expected, IT2FNN-

HOSMC reveals better controlling factors in comparison to SMC. In contrast to conventional SMC, the

combined method is able to reach and track the desired path much faster. By taking a glance at the

position errors curves in Figures 15 and 16, the contrast between the performance of the SMC and the

proposed method is more obvious. According to these figures, the position error of the joints converges to

zero by using SMC much slower than the combined method. Figures 16 and 17 compare chattering effect

for both controllers in both joints. Again here, as soon as SMC reach the sliding surface, the chattering

effect begins to happen whereas the proposed method is able to handle this issue more effectively and

keep the chattering in an acceptable amount (Fig 19).

Figure 12. Random Perturbation Profile



Figure 13. Comparison between SMC and IT2FNN-HOSMC in trajectory tracking of joint 1 under random

perturbation

Figure 14. Comparison between SMC and IT2FNN-HOSMC in trajectory tracking of joint 2 under random

perturbation

Figure 15. Position error in joint 1 by using SMC and IT2FNN-HOSMC under random perturbation



Figure 16. Position error in joint 2 by using SMC and IT2FNN-HOSMC under random perturbation

Figure 17. Chattering effect in joint 1 by using SMC and IT2FNN-HOSMC under random perturbation

Figure 18. Chattering effect in joint 2 by using SMC and IT2FNN-HOSMC under random perturbation



Figure 19. Sliding Variable for random perturbation (IT2FNN-HOSMC)

When more intense perturbation is exerted on the system, SMC is unable to prevent response trajectory to

be shifted into perturbation profile. In general, any control algorithm start imitating the perturbation

profile for its trajectory error diagram from a level which is called Perturbation Resistance Threshold

(PRT). In this paper, IT2FNN-HOSMC is able to increase its PRT significantly in comparison to SMC.

5. FUTURE WORKS AND CONCLUSIONS

In this paper, a hybrid controller is presented for a robotic manipulator with parametric uncertainties under

external perturbations. In this regard, the algorithm is based on interval type 2 fuzzy logic with neural

network core which its inputs are calculated by a higher order sliding surface is presented. Base on

numerical simulations, conventional SMC performance is limited by chattering and delay, but

performance of the proposed algorithm due to using IT2FS (which is able to overcome chattering) and

HOSMC (which is able to reduce delay), is more robust. Although control torque is limited in a range of [-

20, 20] N.m, novel algorithm is able to follow desired trajectory as neural network can satisfy input

constraints. In addition, PRT for IT2FNN-HOSMC performance is higher than SMC which shows the

robustness of the proposed controller. In future work, run time of the algorithm should be improved which

is caused by its nonlinear structure and capability to be updated in time stamps. Neural network parameter

regulation would be applicable idea.

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